The more important identities.
You You don’t have to know all a ll the identities off the top of your head. But these you should.
Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine.
The Pythagorean formula for sines and cosines. This is probably the most important trig identity. Identities expressing trig functions in terms of their complements. There's not much to these. Each of the si trig functions is e!ual to its co"function evaluated at the complementary angle. Periodicity of trig functions. #ine, cosine, secant, and cosecant have period $π $π while while tangent and cotangent have period π . Identities for negative angles. #ine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions.
Ptolemy’s identities, the sum and difference formulas for sine and cosine.
Double angle formulas for sine and cosine. %ote that there are three forms for the double angle formula for cosine. You only need to know one, but be able to derive the other two from the &ythagorean formula.
The less important identities.
You You should know that there are these identities, but they are not as important as those mentioned above. They can all be derived from those above, but sometimes it takes a bit of work to do so.
The Pythagorean formula for tangents and secants. There’s also one for cotangents and cosecants, but as cotangents and cosecants are rarely needed, needed , it’s unnecessary. Identities expressing trig functions in terms of their supplements.
Sum, difference, and double angle formulas for tangent.
The half angle formulas. The ones for sine and cosine take the positive or negative s!uare root depending on the !uadrant of the angle θ $. $. (or eample, if θ $ $ is an acute angle, then the positive root would be used.
Truly obscure identities.
These are )ust here for perversity. %o, not really. They They have some applications, but they’re usually narrow applications, and they could )ust as well be forgotten until needed.
Product-sum identities. This group of identities allow you to change a sum or difference of sines or cosines into a product of sines and cosines.
Product identities. *side+ weirdly enough, these product identities were used before logarithms were invented in order to perform multiplication. ere’s how you could use the second one. -f you want to multiply x multiply x times times y, use y, use a table to look up the angle α whose cosine is x is x and and the angle β angle β whose whose cosine is y is y.. ook up the cosines of the sum α / β / β . and the difference α – β . *verage those two cosines. You get the product xy product xy00 Three table look"ups, and computing a sum, a difference, and an average rather than one multiplication. Tycho Tycho Brahe 12345625728, among others, used this algorithm known as prosthaphaeresis. prosthaphaeresis.
Triple angle formulas. You You can easily reconstruct these from the addition and double angle formulas.
ore half-angle formulas. These describe the basic trig functions in terms of the tangent of half the angle. These are used in calculus for a particular kind of substitution in integrals sometimes called the 9eierstrass t "substituti "substituti
